Chapter 4 Control Charts for Measurements with Subgrouping (for One Variable)

1. Chapter 4Control Charts for Measurements with Subgrouping (for One Variable)

2. 4.9 Determining the Point of a Parameter Change • A change in the mean: The max. likelihood estimator is the max over t of , with T denoting the time of a signal occurs. (“Identifying the time of a step change with X-bar control chart” by Samuel, Pignatiello, and Calvin, QE, 1998) • A change in the variance: (“Identifying the time of a step change in a normal process variance” by Samuel, Pignatiello, and Calvin, QE, 1998)

3. 4.10 Acceptance Sampling and Acceptance Control Chart • Acceptance sampling is not a process control technique. • Acceptance sampling plan specifies the sample size that is to be used and the decision criteria that are to be employed in determining whether a lot or shipment should be rejected. • “You cannot inspect quality into a product” Harold F. Dodge • Studies show that only 80% of non-confirming units are detected during 100% final inspection. • Acceptance sampling should be used only temporarily. • Problems with acceptance sampling plans include the fact that the producer’s risk and the consumer’s risk can both be unacceptably high.

4. 4.10.1 Acceptance Control Chart • Acceptance control limits are determined from the specification limits (Far from 3 range) • APL (Acceptable Process Level): the process level that yields product quality to be accepted 100(1-)% • : Probability of rejecting an APL • RPL (RejectableProcess Level): the process level that yields product quality to be rejected 100(1-)% •  : Probability of accepting an RPL • p1: acceptable % of units falling outside the spec. • p2: rejectable% of units falling outside the spec.

5. 4.10.1 Acceptance Control Chart (4.4) (4.5) (4.6) (4.7)

6. 4.10.1.1 Acceptance Chart with Control Limits • The acceptance chart can be constructed with the -chart control Limits either shown or not shown on the chart. • Not in accordance with contemporary views on quality improvement

7. 4.10.1.1 Acceptance Chart Example • Use data from Table 4.2 • Assume the specification limits are at (USL=150.6495, LSL=-31.7745) • p1 =.00001, =.05 =98.3165 =20.5585 • Recall the -chart UCL=82.2552, LCL=36.6198

8. Table 4.2 Data in Subgroups Obtained at Regular Intervals

9. 4.11 Modified Limits • If the specification limits were at k, the limits would be widened by (k-3)

10. 4.12 Difference Control Charts • The general idea is to separate process instability caused by uncontrollable factors from process instability due to assignable causes. • This is accomplished by taking samples from current production and also samples from what is referred to as a reference lot. • The reference lot consists of units that are produced under controlled process conditions except for possibly being influenced by uncontrollable factors. • Since both samples are equally influenced by uncontrollable factors, any sizable differences between sample means should reflect process instability due to controllable factors.

11. 4.12 Difference Control Charts • The 3-sigma control limits are Where and are avg. range for the reference lot and the current lot • Difference Control Chart replaces -chart, with ( plotted • It is comparable to a pooled-t test for the equality of population means with a significance level of .0027 provided that (is approximately normal distribution.

12. 4.12 R-chart of Difference • The 3-sigma control limits on an R-chart are • ) are plotted

13. 4.12 Paired Difference Control Charts • The 3-sigma control limits are Where R = largest difference between pairs – smallest difference • Difference Control Chart replaces -chart, with ( plotted

14. 4.13 Other Charts • Median chart is a substitute for an -chart. • It is not as efficient as the average • Midrange (average of the largest and the smallest) chart • Coefficient of variation (/) chart

15. 4.14 Average Run Length (ARL) • If the parameters were known, the expected length of time before a point plots outside the control limits could be obtained as the reciprocal of the probability of a single point falling outside the limits when each point is plotted individually. • The expected value is called the Average Run Length (ARL). • It is desirable for the in-control ARL to be reasonably large. • The parameter-change ARL should be small.

16. 4.14 Average Run Length (ARL) • With 3-sigma limits, the in-control ARL is 1/.0027=370.37 • Assume 1 increase in the mean, the parameter-change ARL is 43.89 • When the parameters are estimated, both the in-control ARL and the parameter-change ARL are inflated.

17. 4.14.1 Weakness of the ARL Measure • The run length distribution is quite skewed so that the ARL will not be the typical run length • The standard deviation of the run length is quite large

18. 4.15 Determining the Subgroup Size • By convenience: 4 or 5 • Economic design of control charts • Using graphs (such as given by Dockendorf, 1992) • The larger the subgroup, the more power a control chart will have for detecting parameter changes. • Survey showed most respondents used subgroup size of about 6.

19. 4.15.1 Unequal Subgroup Sizes • May caused by missing data • Minitab (with 2 columns) Variable Sample Size (VSS) • Smaller sample sized is used if the sample mean falls within “warning limits” (2-sigma limits) • Larger sample sized is used if the sample mean falls between warning limits (2-sigma limits) and control limits. • Superior in detecting small parameter changes

20. 4.16 Out-of-Control Action Plans(OCAPs) A flow chart with • Activator: out-of-control signal (limits + run rules) • Checkpoint: potential assignable causes • Terminator: action taken to resolve the condition

21. 4.17 Assumptions for Control Charts • White noise model (when process is in control): • Normality • Independence

22. 4.17.1 Normality • For R-, S-, and S2-charts, the basic assumptions are the individual observations are independent and normally distributed. • The distributions of R, S, and S2 differ considerably from a normal distribution. • Many process characteristics will not be well approximated by normal distribution. (diameter, roundness, mold dimensions, customer waiting time, leakage from a fuel injector, flatness, runout, and percent contamination) • Non-normality is not a serious problem unless there is a considerable deviation from normality.

23. 4.17.2 Independence • The estimation of by is appropriate only when the data are independent. • The appropriate expression for can be determined from the time series model

24. 4.17.2 Example of Invalid Assumption of Independence • First-order autoregressive (AR) process Where and

25. Table 4.6 100 Consecutive Values from AR(1) and

26. Figure 4.7

27. 4.17.2 Remedy for Correlated Data • Fitting a time-series model to the data and using the residuals from the model in monitoring the process • Drawbacks of residual chart: • poor ARL properties • Harder to relate to a residual • Solution: -chart plotted with residual chart

28. 4.18 Measurement Error • Assume measurement variability is independent of product variability, and that repeatability variability and reproducibility variability are independent, then observations = product + repeatability + reproducibility • Reproducibility variability: Determined by the performance of the measurement process under changing conditions (DOE) • Using control chart to determine if reproducibility is in a state of statistical control • Repeatability variability: Estimated using at least a moderately large number of measurements under identical conditions. Var(S2) = 24/(n-1)

29. 4.18.1 Monitoring Measurement Systems • Separate monitoring of variance components for repeatability and reproducibility, or a simultaneous procedure.